K. Takatsuka, NONLINEAR DYNAMICS IN COUPLED FUZZY CONTROL-SYSTEMS .1. COHERENCE ANDCHAOS-FRUSTRATION IN TRIANGLE CONFIGURATION, Physica. D, 82(1-2), 1995, pp. 95-116
Nonlinear dynamics and chaos are studied in a system for which a compl
ete set of equations of motion such as equations of Newton, Navier-Sto
kes and Van der Pol, is not available. As a very general system as suc
h, we consider coupled classical spins (pendulums), each of which is u
nder control by a fuzzy system that is designed to align the spin to a
n unstable fixed point. The fuzzy system provides a deterministic proc
edure to control an object without use of a differential equation. The
positions and velocities of the spins are monitored periodically and
each fuzzy control gives a momentum to its associated spin in the reve
rse directions. If the monitoring is made with an interval short enoug
h, the spin-spin interactions are overwhelmed by the fuzzy control and
the system converges to a state as designed. However, a long-interval
monitoring induces dynamics of ''too-late response'', and thereby res
ults in chaos. A great variety of dynamics are generated under very de
licate balance between the fuzzy control and the spin-spin interaction
, in which two independent mechanisms of creating negative and positiv
e ''Liapunov exponents'' interact with each other.